Review by Stephen Parrott 

				of 
	
		"Decoherence and the Quantum-to-Classical Transition" 

		      by Maximilian Schlosshauer, Springer, 2009.


I obtained this book via interlibrary loan with the intention of buying
it if it looked as if it would repay careful study.  In the three week loan
period I was able to read most of it, though I only skimmed several
chapters on applications.  It is an interesting book with many good features
which I am happy to have read, but I decided not to buy it because I doubt
that it would sufficiently repay careful study.

I am a mathematician with extensive experience in quantum mechanics but little
knowledge of decoherence.  Since my motive was to learn about decoherence,
I will review the book from a student's perspective.

The model of decoherence from which the book draws most of its conclusions 
seems to me unrealistically simplistic.  It is described in such a vague way
that I cannot be sure that I fully understand the author's intent.  The
following should be taken as my best guesses as to that intent.

We start with a quantum system of interest S and 
a given collection of quantum pure states |s0 >, |s1 >, ...  of  S, which are 
usually assumed orthogonal.  (For notational simplicity, we do not assume
them to be normalized.)
The system S is coupled to a quantum "environment" system E, 
which is usually inaccessible to direct observation.
The composite system is mathematically described as the tensor
product of S and E.

Initially, the environment is assumed to be in a "ready" pure state |r >, 
the nature of which is unspecified.   If S is in state |si > (i = 0, 1, ...), 
the composite system is initially in the product state  |si > |r >.  
It is *assumed* that such a state will evolve after a certain time 
to another product state |si > |ei >, where
the |ei > are pure states of the environment, depending on i (and also on 
time, which is not included in the present notation for simplicity).  The
environmental states |ei > are usually assumed orthogonal, or approximately so.
Under these assumptions, a superposition like (|s0 > + |s1 >)|r > 
will evolve to the state |s0 > |e0 > + |s1 >|e1 >, 
which is generally an entangled
state of the composite system (i.e., it is not a product state).  
According to most formulations of quantum mechanics, 
the state of S  is then the partial trace of (the projector on) that entangled state.  
This partial trace is generally a mixed (not pure) state.  
(The book takes the unusual position that
S has no state when the composite system is in that entangled state, but
reaches the same conclusions as if the state of S were the just-mentioned
partial trace.)  This conversion of the superposition |s0 > + |s1 > (a pure
state) into a non-pure mixed state seems to be 
what the book calls "decoherence" or a "quantum-to-classical transition". 

Classical statistical mechanics describes states as mixed states, but they
are mixtures of classical probability distributions, not mixtures of quantum
pure states as is the partial trace of |s0>|e0> + |s1>|e1>, so identifying
that partial trace as a "classical" state would seem 
to require further justification.   Why can't a mixture 
of pure *quantum* states exhibit non-classical properties? 
This point seems to be overlooked in the book, 
which seems to implicitly identify the mixed
quantum state just mentioned with a classical state, describing the process
(|s0> + |s1>)|r> ---> |s0>|e0> + |s1>|e1> as a transition from the quantum
realm to everyday classical physics.

The process just described is my best guess at what  
what the book means by "decoherence".
I was several hundred pages into the book 
when I realized that I still wasn't sure precisely what the book *did* mean by decoherence".  
So, I reread the introductory chapters, 
which had seemed to only partially make sense on first reading.  
They didn't make any more sense on second 
reading.  My many initial questions remained.

The above brief description is intended to give a sense of the basic 
assumptions on which the book builds.     If they seem to you sufficiently
compelling to support 400 pages of reasoning (they didn't to me), 
then you will probably like the book 
because it is generally well-written and well-edited.

It is written in a pedagogical style which uses simple examples to 
make its points.  It is obvious that both the author and publisher 
have taken great pains with its presentation.  

"Decoherence" phenomena have only recently become
experimentally accessible, so the book describes a real frontier of knowledge.
A few beautiful experiments have been done and are described unusually 
well, with excellent diagrams.  In several instances it is stated that the
experimental results agree well with theoretical predictions, but  
these theoretical predictions are not worked out in detail in the book, 
so the interested reader will have to go to the original literature to find out 
what assumptions and approximations were made and to judge for himself 
the extent to which the experiments confirm the theory. 

This is a wordy book, which in parts seems more akin to a philosophy tome than
a physics text.  There are whole sections without any equations, and 
I often found it hard to understand precisely what points the author was
trying to make. 

I noticed only a few mathematical errors in the book, and 
it seems unusually typo-free.  Its editing and diagrams are outstanding.  
It includes an extensive bibliography with over 500 references, 
and the text makes clear that the author has actually read many of them. 
If I could have agreed with its basic premises and followed its logic in 
detail, I would probably have considered it a great book.  

March 6, 2009